We have a small SMP server in Minecraft and I set up an iron golem farm
according to

(only redstone in the hopper sorter system) and an automatic wheat farm according to

(minecart running beneath the farm and a redstone circuit for the unloading as well as a hopper sorting system). Both farms work very well as long as a player is online.

Of course, they stop working after the somewhat 60s after the last player has logged out. So I went and read a bit about that topic and came up with the idea to send an item through a nether portal every 45s via a piston/hopper clock plus a dropper. This kinda seems to work, since my iron golem farm is drowning itself in iron. However, the wheat farm seems to stop working when no one is online, as well as any minecart/redstone builds in the spawn chunks. Is there any mechanic at work here that I overlooked?

Spawn chunks will not be unloaded from memory, regardless of how far away a player moves, so long as there is at least one player in the
overworld.All overworld chunks (including the spawn chunks) are
unloaded 60 seconds after the last player in the overworld enters the
Nether or the End.

In your case, I believe that the chunks do not get unloaded, due to your sending an item through the nether portal(items on the ground are considered entities).

Unfortunately, spawn chunks are only loaded if there is at least one player on the server. You could have an unused account permanently in /gamemode 3 on the server near the chunks if you want. That way they would nevver unload. If you have an AFK kick on your server you could make a small AFK machine See this video, and have an unused player account permanently in /gamemode 1 constantly cycling this machine near the farms. This will also keep the chunks loaded.

yes, this is what I thought, however the fact that the iron farm keeps on working even when I know for sure that no one was online for at least a few days is speaking somewhat against that... the same goes for the automatic chicken farm, which also keeps producing when no one is online. There has to be more to that story.
– Prof. FarnsworthOct 26 '15 at 14:33

@Prof.Farnsworth The spawn chunks don't truly "unload" when no players are online. What happens is entities and tile entities will no longer be sent an update tick, but everything beyond that will still work. Villages will still spawn iron golem farms, for example, but those iron golems will not move due to not receiving an update tick. If you have a chunk loader in place that is capable of loading enough chunks to tick entities, that would explain why that works.
– SkylinerwOct 27 '15 at 10:12

@Skylinerw Well, I used this concept from here youtube.com/watch?v=dkxRvYKwNZU . But then I am still confused. If the golems, and obviously the chicken as well, do receive an update tick as entities, why don't the minecarts and the villager in the farm receive one? (everything is inside the spawn chunk area)
– Prof. FarnsworthOct 27 '15 at 13:05